Sharp Hessian integrability estimates for nonlinear elliptic equations: An asymptotic approach

We establish sharp W-2,W-p regularity estimates for viscosity solutions of fully nonlinear elliptic equations under minimal, asymptotic assumptions on the governing operator F. By means of geometric tangential methods, we show that if the recession of the operator F - formally given by F{*} (M) := infinity(-1) F(infinity M) - is convex, then any viscosity solution to the original equation F(D(2)u) = f(x) is locally of class W-,(2,p) provided f is an element of L-P, p > d, with appropriate universal estimates. Our result extends to operators with variable coefficients and in this setting they are new even under convexity of the frozen coefficient operator, M bar right arrow F(x(0), M), as oscillation is measured only at the recession level. The methods further yield BMO regularity of the Hessian, provided the source lies in that space. As a final application, we establish the density of W-2,W-p solutions within the class of all continuous viscosity solutions, for generic fully nonlinear operators F. This result gives an alternative tool for treating common issues often faced in the theory of viscosity solutions. (C) 2016 Elsevier Masson SAS. All rights reserved. (AU)